A multigrid method for predicting periodically fully developed flow

The use of multigrid methods in complex fluid flow problems is still under development. In this paper a full multigrid procedure has been incorporated in a finite volume solution for predicting fully developed fluid flow in a streamwise periodic geometry. Steady computations in two-dimensional body fitted co-ordinates have shown considerable savings in computation time by this multigrid method.     

A multigrid method for steady incompressible navier-stokes equations based on flux difference splitting

The steady Navier–Stokes equations in primitive variables are discretized in conservative form by a vertex-centred finite volume method Flux difference splitting is applied to the convective part to obtain an upwind discretization. The diffusive part is discretized in the central way. In its first-order formulation, flux difference splitting leads to a discretization of so-called vector positive type. This allows the use of classical relaxation methods in collective form. An alternating line Gauss–Seidel relaxation method is chosen here. This relaxation method is used as a smoother in a multigrid method. The components of this multigrid method are: full approximation scheme with F-cycles, bilinear prolongation, full weighting for residual restriction and injection of grid functions. Higher-order accuracy is achieved by the flux extrapolation method. In this approach the first-order convective fluxes are modified by adding second-order corrections involving flux limiting. Here the simple MinMod limiter is chosen. In the multigrid formulation the second-order discrete system is solved by defect correction. Computational results are shown for the well known GAMM backward-facing step problem and for a channel with a half-circular obstruction.     

Finite element solutions for laminar and turbulent compressible flow

The time-split finite element method is extended to compute laminar and turbulent flows with and without separation. The examples considered are the flows past trailing edges of a flat plate and a backward-facing step. Eddy viscosity models are used to represent effects of turbulence. It is found that the time-split method produces results in agreement with previous experimental and computational results. The eddy viscosity models employed are found to give accurate predictions in all regions of flow except downstream of reattachment.     

A multigrid method for steady incompressible Navier–Stokes equations based on partial flux splitting

Flux splitting is applied to the convective part of the steady Navier–Stokes equations for incompressible flow. Partial upwind differences are introduced in the split first-order part, while central differences are used in the second-order part. The discrete set of equations obtained is positive, so that it can be solved by collective variants of relaxation methods. The partial upwinding is optimized in the same way as for a scalar convection–diffusion equation, but involving several Peclet numbers. It is shown that with the optimum partial upwinding accurate results can be obtained. A full multigrid method in W-cycle form, using red–black successive under-relaxation, injection and bilinear interpolation, is described. The efficiency of this method is demonstrated.     

Improvement of a pressure gradient method and its application to an unsteady flow problem

The pressure gradient method using velocity components and components of a pressure gradient as dependent variables has been modified to solve incompressible Newtonian fluid flow problems numerically. Applying this modified method to unsteady-state development of flow in a circular cavity shows that, at least for the case of a low Reynolds number flow, relative errors produced by the proposed method are smaller for most time intervals than those produced by the primitive velocity-pressure variable method and by the standard pressure gradient method. Also it is found that the modified and standard pressure gradient methods can be applied to the unsteady circular cavity flow at a moderate Reynolds number of at least up to 200.     

Computation of fluid flow with a parallel multigrid solver

A finite volume numerical method for the prediction of fluid flow and heat transfer in simple geometries was parallelized using a domain decomposition approach. The method is implicit, uses a colocated arrangement of variables and is based on the SIMPLE algorithm for pressure-velocity coupling. Discretization is based on second-order central difference approximations. The algebraic equation systems are solved by the ILU method of Stone.¹ To accelerate the convergence, a multigrid technique was used. The efficiency was examined on three different parallel computers for laminar flow in a pipe with an orifice and natural convection in a closed cavity. It is shown that the total efficiency is made up of three major factors: numerical efficiency, parallel efficiency and load-balancing efficiency. The first two factors were thoroughly investigated, and a model for predicting the parallel efficiency on various computers is presented. Test calculations indicate reasonable total efficiency and favourable dependence on grid size and the number of processors.     

Extension of multigrid methodology to supersonic/hypersonic 3-D viscous flows

A multigrid acceleration technique developed for solving the three-dimensional Navier–Stokes equations for subsonic/transonic flows has been extended to supersonic/hypersonic flows. An explicit multistage Runge–Kutta type of time-stepping scheme is used as the basic algorithm in conjunction with the multigrid scheme. Solutions have been obtained for a blunt conical frustum at Mach 6 to demonstrate the applicability of the multigrid scheme to high-speed flows. Computations have also been performed for a generic High-Speed Civil Transport configuration designed to cruise at Mach 3. These solutions demonstrate both the efficiency and accuracy of the present scheme for computing high-speed viscous flows over configurations of practical interest.     

A residual method of finite differencing for the elliptic transport problem and its application to cavity flow

A residual method of finite differencing the governing differential equation for the elliptic transport problem is presented. The new finite differencing technique is applied to (1) the one-dimensional transport problem and (2) the cavity flow problem for numerical illustrations. The results indicate the validity of the residual method of finite differencing. The usual method of term-by-term finite differencing, and considerations such as central differencing, hybrid differencing and upwind differencing are not needed in the present residual method.     

Overlapping grids and multigrid methods for three-dimensional unsteady flow calculations in IC engines

A new computational methodology with emphasis on using an overlapping grid technique and a multigrid method has been developed. The main feature of the present overlapping-grid system is of extended flexibility to deal with three-dimensional complex multicomponent geometries. The multigrid method is incorporated into this technique to accelerate the convergence of the numerical solution. The current scheme has been applied for computations of the laminar flows in the multicomponent configuration of internal combusion engines. The flow is governed by three-dimensional, time-dependent, incompressible Navier-Stokes equations with the continuity equation. A time-independent grid system is constructed for the moving boundary, i.e. the moving piston in the engine. This grid system is entirely different from others for the same problem in previous works. The performance of the present method has been validated by comparing the results with those from an equivalent, single-grid method and those from experiments. In addition, the flexibility and potential of the method has been demonstrated by calculating several cases which would be very difficult to be handled by other schemes.     

Finite element analysis of density flow using the velocity correction method

A finite element method is proposed for the analysis of density flow which is induced by a difference of density. The method employs the idea that density variation can be pursued by using markers distributed in the flow field. For the numerical integration scheme, the velocity correction method is successfully used, introducing a potential for the correction of velocity. This method is useful because one can use linear interpolation functions for velocity, pressure and potential based on the triangular finite element. The final equations can be formulated using the quasi-explicit finite element method. A flume in a tank with sloping bottom has been analysed by the present method. The computed results show extremely good agreement with the experimental observations.     

Application of the Dorodnitsyn finite element method to swirling boundary layer flow

The Dorodnitsyn finite element method for turbulent boundary layer flow with surface mass transfer is extended to include axisymmetric swirling internal boundary layer flow. Turbulence effects are represented by the two-layer eddy viscosity model of Cebeci and Smith¹ with extensions to allow for the effect of swirl. The method is applied to duct entry flow and a 10 degree included-angle conical diffuser, and produces results in close agreement with experimental measurements with only 11 grid points across the boundary layer. The introduction of swirl (w_e/u_e = 0.4) is found to have little effect on the axial skin friction in either a slightly favourable or adverse pressure gradient, but does cause an increase in the displacement area for an adverse pressure gradient. Surface mass transfer (blowing or suction) causes a substantial reduction (blowing) in axial skin friction and an increase in the displacement area. Both suction and the adverse pressure gradient have little influence on the circumferential velocity and shear stress components. Consequently in an adverse pressure gradient the flow direction adjacent to the wall is expected to approach the circumferential direction at some downstream location.     

Multilevel methods for temporal and spatial flow transition simulation in a rough channel

We investigate the instability of 2D incompressible flows in a rough planar channel by tracking the growth of the unstable mode in its early stage. We develop both second- and fourth-order finite difference methods on a staggered grid, together with a fully implicit time-marching scheme, using grid generation to accommodate fairly general geometries. A multigrid full approximation scheme based on the line-distributive relaxation method is used for fast convergence. For a 2D smooth channel, numerical results show good agreement with the analytic solution obtained from linear theory for small disturbances. Numerical results for a 2D channel with one and two roughness elements are analysed by Fourier analysis. They show how the roughness elements affect the growth of the perturbation.     

避免核函数导数的二阶导数核近似方法的修正

采用核近似光滑粒子流体力学SPH（Smooth Particle Hydrodynamics）方法计算一元函数二阶导数和多元函数二阶偏导数,对避免核函数导数算法进行了改进,提出了对核函数光滑长度处处具有o（h）精度的算法,并分别推导出一元和多元函数的修正公式。用不同的粒子间距和不同的光滑长度进行了二阶导数的计算和热传导问题的模拟,进行了修正方法与原方法的误差分析。结果表明,本文提出的修正措施在提高精度、减少误差及加快收敛速度等方面起到很大的作用。     

Variable penalty method for finite element analysis of incompressible flow

A new scheme is applied for increasing the accuracy of the penalty finite element method for incompressible flow by systematically varying from element to element the sign and magnitude of the penalty parameter λ, which enters through ?.v + p/λ = 0, an approximation to the incompressibility constraint. Not only is the error in this approximation reduced beyond that achievable with a constant λ, but also digital truncation error is lowered when it is aggravated by large variations in element size, a critical problem when the discretization must resolve thin boundary layers. The magnitude of the penalty parameter can be chosen smaller than when λ is constant, which also reduces digital truncation error; hence a shorter word-length computer is more likely to succeed. Error estimates of the method are reviewed. Boundary conditions which circumvent the hazards of aphysical pressure modes are catalogued for the finite element basis set chosen here. In order to compare performance, the variable penalty method is pitted against the conventional penalty method with constant λ in several Stokes flow case studies.     

Use of a three-component model to compute gas suspension flow and rarefied flow over bodies

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 39–42, January–February, 1991.     

代数多重网格方法在紊流数值预测中的应用

机械和矿山工程中广泛使用锥形渐扩管。将DLR型k-ε紊流模型中非线性偏微分方程基于全隐式高精度迎风差分格式离散,得到差分方程的系数矩阵为五对角块十三对角带状稀疏矩阵,基于一种"三元组"方式进行压缩存储,节约内存。提出了一种基于DLR型k-ε紊流模型与代数多重网格方法结合的新算法,阐述了代数多重网格方法的实施过程。对具有逆压梯度流动的锥形渐扩管内紊流进行了数值预测。数值实验表明,代数多重网格方法对求解紊流模型离散方程组非常有效,同此前该紊流数值模拟中使用的Point-SOR方法相比,计算效率有了显著提高,计算结果与实验结果吻合较好。     

Two-step explicit finite element method for two-layer flow analysis

A finite element method for the analysis of two-layer density flows is presented in this paper. The standard Galerkin method based on linear interpolation functions is used to yield discrete spatial variables. For numerical integration in time, an explicit two-step selective lumping method is used. Here it is applied to a flow analysis of Ishikari Bay, at the mouth of Ishikari River. This case demonstrates a procedure that yields a numerically stable solution.     

Selective lumping finite element method for shallow water flow

A finite element method for solving shallow water flow problems is presented. The standard Galerkin method is employed for spatial discretization. The numerical integration scheme for the time variation is the explicit two step scheme, which was originated by the authors and their co-workers. However, the original scheme has been improved to remove the erroneous artifical damping effect. Since the improved scheme employs a combination of lumped and unlumped coefficients, the scheme is referred to as a selective lumping scheme. Stability conditions and accuracy are investigated by considering several numerical examples. The method has been applied to the tidal flow in Osaka Bay and Yatsushiro Bay.     

Symmetric Gauss-Seidel multigrid solution of the Euler equations on structured and unstructured grids

The efficient symmetric Gauss-Seidel (SGS) algorithm for solving the Euler equations of inviscid, compressible flow on structured grids, developed in collaboration with Jameson of Stanford University, is extended to unstructured grids. The algorithm uses a nonlinear formulation of an SGS solver, implemented within the framework of multigrid. The earlier form of the algorithm used the natural (lexicographic) ordering of the mesh cells available on structured grids for the SGS sweeps, but a number of features of the method that are believed to contribute to its success can also be implemented for computations on unstructured grids. The present paper reviews, the features of the SGS multigrid solver for structured gr0ids, including its nonlinear implementation, its use of “absolute” Jacobian matrix preconditioning, and its incorporation of multigrid, and then describes the incorporation of these features into an algorithm suitable for computations on unstructured grids. The implementation on unstructured grids is based on the agglomerated multigrid method developed by Sørensen, which uses an explicit Runge-Kutta smoothing algorithm. Results of computations for steady, transonic flows past two-dimensional airfoils are presented, and the efficiency of the method is evaluated for computations on both structured and unstructured meshes.